Question about causality in CDT

[nightcleaner]

Causal Dynamic Triangulation starts out with the premise that causality is preserved, even elemental to the theory, but I have not yet understood what makes CDT causal, aside from the title. Where do I look for the cause and effect sequence in CDT?

Thanks,

nc

 

[Hans de Vries]

Causal Dynamic Triangulation starts out with the premise that causality is preserved, even elemental to the theory, but I have not yet understood what makes CDT causal, aside from the title. Where do I look for the cause and effect sequence in CDT?

Thanks,

nc

I would guess it's simply that you don't go back in time. This then should
'renormalize' the theory for 1+1 dimension. It would take light infinitely long
to wiggle through an infinitely curved space-time but by requiring that time
goes only one direction you achieve that the path-length becomes finite.

For instance: If you would allow only positive dt and dx then the total path
length becomes maximal $\Delta x+\Delta y$ ( minimal $\sqrt{\Delta x^2+\Delta y^2}$).

If you allow positive and negative dx and positive only dt then the path
can get infinite only in an artificial case: That is, if for each move dt the
path moves straight infinitely long left and right on de x-axis. Now if you
have a Monte Carlo simulation that determines the curvature then this
will never happen.

It seems to me that the problems start if you want to do the same for 2+1
and 3+1 dimensions where you need additional constraints to renormalize.


Regards, Hans

 

[marcus]

[EDIT oops, didnt see Hans already responded. hope our replies complement each other ]

... what makes CDT causal...

there are several ways to go about explaining why that term was chosen.

as background, keep in mind that before 1998 there was an approach called DT (dynamical triangulations) which modeled spacetime by gluing together simplexes that were chunks of EUCLIDEAN 4-space.

Loll's innovation involved making several changes, an important one of which was to keep on doing DT but to use simplexes that were chunks of LORENTZIAN 4-space.

The simplexes were also to be organized in layers, but for the moment just look at the fact that they were pieces of so-called "Minkowski space", the Lorentzian 4D space of special relativity, rather than Euclidean 4-space.

A nice feature of Lorentzian 4-space is that it has past and future LIGHTCONES defined at every point. The event that I am sitting here typing the letter E has a backwards lightcone and everything inside that cone could have AFFECTED this event.

Anything outside that lightcone could not have influenced that event because information (the 'substance of causality' one might say) cannot travel faster than light.

what I am explaining is that LORENTZIAN 4-SPACE HAS A BUILT-IN CAUSAL STRUCTURE

So with the middle finger of my left hand I type the letter E, and that event could have an influence on all events which are inside the future lightcone of that event E. that event could not influence anything outside the cone because information cannot travel faster than light.

A big feature of the Lorentz transformations on which Spec Rel is based is that they preserve lightcones: the speed of light which determines the angle that the cones spread out is the same for all observers. So all observers can look at an event and agree what set of points is in the future lightcone of a that event----even if they disagree about some other things---and agree on the past lightcone.

So the Lorentz transformations that take you from one observer's frame to another's frame PRESERVE THE CAUSAL STRUCTURE of the 4D space. Everyone can agree on what events might conceivably have influenced me to type the letter E---events in my down-to-earth past as well as events in distant galaxies whose light reached the Earth before I typed it.

Practically speaking, as I look at it, it would be silly to say that the fart of a green lemming on Andromeda could be causally connected to my typing this letter. But no matter how silly it is, if it is an event in my past lightcone then in some bizarre way the influence of that event could have traveled to me and affected this event here and now. So this possible connection is part of the CAUSAL STRUCTURE-----a web of conceivable, if generally impractical, farfetched and ridiculous, influences.

The lightcone causal structure buit-in to Lorentzian 4-space is such an important feature of it that often people will use the word "causal" virtually as a synonym for "Lorentzian".
The causal structure is the primary feature which distinguishes Lorentzian 4-space from Euclidian 4-space (which does not have lightcones that different observers can agree on).

In 1998 when Ambjorn and Loll took ordinary DT and decided to try using simplexes made of Lorentzian 4-space instead of Euclidean, from that point on they called the old approach EDT EUCLIDEAN DYNAMICAL TRIANGULATIONS. And their new approach they called LDT LORENTZIAN DYNAMICAL TRIANGULATIONS.

Later on, they observed that "causal" has 2 syllables and is easier to say than "Lorentzian" which has 4 syllables. So they streamlined the terminology by making it CDT. This is not only easier to say but also highlights causal structure, an important distinguishing feature.

But Loll still does sometimes use the older term "Lorentzian dynamical triangulation" and indeed employed it in the abstract of a paper as recent as June 2005!
You will see this term in the brief summary of her Black Hole paper with Dittrich, given at
http://arxiv.org/gr-qc/0506035

 

[nightcleaner]

"Thanks for the replies, I will take them offline for study. Still haven't got my 'phone line in at the new place, but hope is for next week. Meanwhile, have limited time, so will read and reply offline.

Thanks again,

nc